# Modelling Regime Changes of Dunes to Upper-Stage Plane Bed in Flumes and in Rivers

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Dune Model

#### 2.1. General Set-Up

#### 2.2. Flow Model

_{v}denotes the constant vertical eddy viscosity. Note that a steady flow model is used to compute unsteady flow of a flood wave. This is a reasonable approach, because the length scale of a dune is small, so the remaining terms are large compared to the time derivatives. The computational domain is shown in Figure 2.

_{b}is the bed level relative to the x-axis (in m). The flow is forced in the domain because the x-axis is actually at a slope i with regard to the real horizontal plane, creating a water level difference along the domain.

#### Boundary Conditions

_{b}). The boundary conditions at the water surface, Equation (3) represents no flow through the surface, and Equation (4) means no shear stress at the surface (so wind stress is assumed to be negligible). The kinematic boundary condition at the bed, Equation (5) yields that there is no flow through the bed.

_{b}(m

^{2}/s

^{2}) represents the volumetric bed shear stress (i.e., without the density), u

_{b}(m/s) is the flow velocity along the bed, and the resistance parameter S (m/s) controls the resistance at the bed. For more details about the model equations and numerical solution procedure, reference is made to Paarlberg et al. (2009) [21] and Van den Berg et al. (2012) [35]. For more details, the reader is referred to van Duin et al. (2016) [26].

#### 2.3. Bed Load Sediment Transport Model

^{−1}) is determined by:

_{0}= 0.03, θ is the Shields parameter, θ

_{c}is the critical Shields parameter, and Δ = ρ

_{s}/ρ−1. The grain density ρ

_{s}is set to 2650 kg/m

^{3}, and the density of the water ρ is set to 1000 kg/m

^{3}

_{c}(x), corrected for bed slope effects, is given by the following equation:

_{c0}the critical volumetric bed shear stress for flat bed, defined by Equation (9) and η = tan(φ)

^{−1}, in which the angle of repose φ = 30° for sand. In this equation, θ

_{c0}is the critical Shields parameter for flat bed, and D

_{50}is the median grain size.

#### 2.4. Step Length

_{*}/w

_{s}(in which u

_{*}= friction velocity and (u

_{*}= (τ/ρ)

^{1/2}), and w

_{s}= settling velocity) from about 0.18 to 0.35. From this data different step length models are derived by various authors. In Section 3, two step length models based on bed load from the literature are implemented, and results are discussed. Hereafter, Section 4 presents a third method in which, in addition to the models of Section 3, suspended load processes are also explicitly accounted for. In general, the step length is assumed to be constant along the dune, so it can vary only over time due to variation in dune-averaged flow parameters.

#### 2.5. Bed Evolution

_{p}= 0.4 is the bed porosity.

## 3. Step Length Models from Literature

#### 3.1. Sekine and Kikkawa (1992) Step Length Models

_{*}(u

_{*}= (τ/ρ)

^{1/2}), and (3) that it is inversely proportional with the settling velocity w

_{s}. The suspension parameter u

_{*}/w

_{s}ranges from about 0.15 to 0.28 in this set of calculations, so bed-load conditions were present (u

_{*}/w

_{s}< 1). The relation between these parameters and the non-dimensional step length α is as follows:

_{2}= 3.0 × 10

^{3}, and u

_{*c}is the critical friction velocity (u

_{*c}= ${\left({\tau}_{c}/\rho \right)}^{1/2}$; note that this is volumetric bed shear stress, which has a unit of m

^{2}/s

^{2}).

#### 3.2. Shimizu et al. (2009) Step Length Models

_{min}= 50) and maximum (α

_{max}= 250) value of non-dimensional step length α in a conceptually derived relation between α and dimensionless grain shear stress θ′. For values of θ′ between 0 and 0.5 (the dune regime), α was assumed to be constant at the minimum value (α

_{min}). For values of θ′ above 0.8 (the upper-stage plane bed regime), α was assumed to be at the maximum value (α

_{max}). In the transitional regime (θ′ from 0.5 to 0.8), α was linearly interpolated. Besides the Shields parameter, there is no further dependency on sediment parameters.

#### 3.3. Comparison of Step Length Models

_{50}of 0.28 mm, a slope i of 2 × 10

^{−3}, and a hydrograph as presented below are used.

## 4. New Step Length Model

_{*}/w

_{s}> 1 for most of the scenario (it varied between 0.9 and 1.8), the parameter α needs to become much higher with the new step length model to implicitly take into account the effects of suspended transport. Suspended sediment load differs from bed load in the sense that turbulent vortices have the potential to move the sediment to higher parts in the water column. Due to the vertical mixing and settling process of suspended sediment, the suspended sediment load does not respond to variations in bed-shear stress (and thereby sediment pick-up) immediately, but with a spatial (and/or time) lag. To some extent, this is similar as bed-load, which also experiences a (smaller) spatial lag due to the step length of sediment grains. The turbulent mixing capacity ε

_{s}is related to eddy viscosity ν

_{t}, which for wall boundary layer flows is related to the friction velocity u

_{*}and the water depth h as follows (see e.g., Van Rijn (1993) [43]):

_{s}, and the settling by gravity (settling velocity w

_{s}) and scales with ε

_{s}/w

_{s}= u

_{*}h/w

_{s}(Rouse, 1937) [44]. Galappatti and Vreugdenhil (1985) [45] derived a depth-averaged model for suspended sediment based on the usual advection-diffusion equation. They show that the vertical processes of turbulent diffusion and settling of suspended sediment can be translated into a relaxation or adjustment process of suspended sediment in flow direction, involving an adjustment length (or spatial lag) of suspended sediment. A generalized expression is as follows:

_{*}, the settling process by the settling velocity w

_{s}.

_{ref}is a reference water depth equal to the water depth at the start of the transitional regime of the case used to tune the step length model, scenario A4 of Shimizu et al. (2009) [14], of which the results can been seen in Figure 5. The value of the non-dimensional grain shear stress-dependent step length α

_{g}follows from a modified version of the Shimizu et al. (2009) [14] step length model, as can be seen in Figure 5. To reiterate, with this modification the step length no longer depends on only bed shear stress or friction velocity as with bed load, but also on the water depth as with suspended transport. It represents processes inherent in the turbulent mixing of suspended material—namely, that larger water depth leads to larger turbulent vortices, which in turn leads to sediment higher in the water column, and due to the size of the vortices, a larger settling distance.

_{min}at θ′ = 0.5 and α

_{max}at θ′ = 0.8 have been tested, and using α

_{min}= 50 and α

_{max}= 350 works best within the new dune evolution model compared to the dune evolution results of Shimizu et al. (2009) [14]. The water depth at the start of the transitional regime was 0.1166 m, which will be used as the reference water depth h

_{ref}. In Figure 5 the currently used model for α

_{g}is compared with the Shimizu et al. (2009) [14] step length model for α

_{S}. It should be noted that this figure is without the additional influence of the water depth as defined in Equation (18). Combining that equation with the figure above and all other previous considerations leads to the following new equation for α:

_{min}= 50, α

_{max}= 350, and h

_{ref}= 0.1166 m.

## 5. Results with Flume Conditions

_{*}/w

_{s}varied between 0.9 and 1.8 during the scenario, so with this sediment and this flow regime the suspension regime is present for most of the time. Using scenario A4 from Shimizu et al. (2009) [14], the following development of the dune field over time is found with the dune evolution model as developed in the present study.

^{2}/s, the water depth in the rising part is clearly higher than in the falling part. In the rising limb, the dunes have had a longer time to grow than in the falling limb, and are therefore higher. Because the dunes are higher, the water depth is higher despite the discharge being the same. Shimizu et al. (2009) [14] have clearly shown this hysteresis effect as well, though for their model it’s more pronounced. They have also reported in the order of 25% lower water depths than we found.

## 6. Results with River Conditions

^{3}/s in the river Waal [47]. For simplicity, the dune evolution model is applied to only the main channel; interactions with the floodplains are not taken into account here. It is assumed that 60% of the peak discharge went through the main channel. The main channel is assumed to be 300 m wide, which corresponds well to the width of the main channel along the river Waal in the SOBEK model made by Deltares and used by Paarlberg (2012) [48]. This would make the specific discharge 12.5 m

^{2}/s, which is rounded to 13 m

^{2}/s for the current study. To simulate these conditions, a D

_{50}of 1.2 mm (in the range presented by Giri et al., 2008 [49]), a slope i of 1 × 10

^{−4}[50], and a hydrograph as presented in Figure 10 are used as input for the model. The settling velocity corresponding to the used D

_{50}is w

_{s}= 0.13 m/s. This hydrograph leads to water depths between 8.5 and 11.1 m, which correspond well to the typical water depths in the Waal for the flood of 1998 as reported by Julien et al. (2002) [47].

#### 6.1. Results for the River Scenario

#### 6.2. Results for the More Extreme River Scenario

## 7. Discussion

_{max}= 300 as well as α

_{max}= 350 are shown in Figure 22. Here it can be seen that the bed is washed out at about 37 min with α

_{max}= 300 instead of 35 min as with α

_{max}= 350, and that the maximum dune height is 3.75 cm and 2.8 cm, respectively. This shows that the results are sensitive to the settings of the step length model, though there is no extreme effect on general model behaviour.

_{*}/w

_{s}and increases with the water depth h (see Equation (17)). Now, by incorporation of the depth in the step length formulation, we allow for much larger steps. Such larger steps can be thought of as overflowing the flow separation zone in a dune, so that the sediment accumulates at the stoss side of the next dune. If this step becomes very large, suspended sediment will be able to pass even more than one dune crest.

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**Hydrograph of scenario A4 after Shimizu et al. (2009) [14].

**Figure 5.**The non-dimensional step length of the non-dimensional step length model of Shimizu et al. (2009) [14], using α

_{S}for bed load, and the currently used step length model Equation (19), using α

_{g}, which represents the grain-shear-stress-dependent part of non-dimensional step length in the new model. In this figure the depth is assumed to be equal to h

_{ref}in order to more clearly show the relation with the Shimizu step length model.

**Figure 7.**Dune crest and trough position (black lines) and water depth (blue line) over time, using scenario A4 from Shimizu et al. (2009).

**Figure 8.**Specific discharge versus non-dimensional step length α separated for the rising and falling stages of the hydrograph.

**Figure 9.**Specific discharge versus water depth separated for the rising and falling stages of the hydrograph, using scenario A4 from Shimizu et al. (2009).

**Figure 12.**Dune crest and trough position (black lines) and water depth (blue line) over time, using the river scenario.

**Figure 13.**Discharge versus dune height for the rising and falling stages of the hydrograph, using the river scenario. The arrow signifies the direction of development over time.

**Figure 14.**Non-dimensional shear stress θ versus dune height for the rising and falling stages of the hydrograph, using the river scenario. The arrow signifies the direction of development over time.

**Figure 15.**Specific discharge versus non-dimensional step length α separated for the rising and falling stages of the hydrograph, using the river scenario. The arrow signifies the direction of development over time.

**Figure 16.**Specific discharge versus water depth separated for the rising and falling stages of the hydrograph, using the river scenario. The arrow signifies the direction of development over time.

**Figure 17.**Dune field over time, using the more extreme river scenario. The right part zooms in on the dune field in the period where the transition occurs.

**Figure 18.**Dune crest and trough position (black lines) and water depth (blue line) over time, using the more extreme river scenario.

**Figure 19.**Discharge versus dune height for the rising and falling stages of the hydrograph, using the more extreme river scenario. The arrow signifies the direction of development over time.

**Figure 20.**Specific discharge versus non-dimensional step length α separated for the rising and falling stages of the hydrograph, using the more extreme river scenario. The arrow signifies the direction of development over time.

**Figure 21.**Specific discharge versus water depth separated for the rising and falling stages of the hydrograph, using the more extreme river scenario.

**Figure 22.**Dune crest and trough position for the first 45 min of scenario A4 of Shimizu et al. (2009) [14]) with (

**a**) α

_{max}= 300 and (

**b**) α

_{max}= 350.

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Duin, O.J.M.v.; Hulscher, S.J.M.H.; Ribberink, J.S.
Modelling Regime Changes of Dunes to Upper-Stage Plane Bed in Flumes and in Rivers. *Appl. Sci.* **2021**, *11*, 11212.
https://doi.org/10.3390/app112311212

**AMA Style**

Duin OJMv, Hulscher SJMH, Ribberink JS.
Modelling Regime Changes of Dunes to Upper-Stage Plane Bed in Flumes and in Rivers. *Applied Sciences*. 2021; 11(23):11212.
https://doi.org/10.3390/app112311212

**Chicago/Turabian Style**

Duin, Olav J. M. van, Suzanne J. M. H. Hulscher, and Jan S. Ribberink.
2021. "Modelling Regime Changes of Dunes to Upper-Stage Plane Bed in Flumes and in Rivers" *Applied Sciences* 11, no. 23: 11212.
https://doi.org/10.3390/app112311212